Closed-form solution for an integral

Question

Is there a closed form solution for the following integral

$$\int_0^z\frac{1}{1+z-x}\frac{1}{(1+x)^K}\,dx$$

for a positive integer $K\geq 1$, and positive real number $z\geq 0$? I searched the table of integrals, but couldn't find something similar.

Answer 1

Hint:

Let $$ I(k)=\frac {1}{z+2} \int_0^z \frac {(1+x)+(1+z-x)}{(1+z-x)(1+x)^k} dx $$

$$I(k)=\frac {1}{z+2} \int_0^z \frac{1}{(1+z-x)(1+x)^{k-1}}+\frac {1}{z+2}\int_0^z (1+x)^{-k}$$

Therefore $$I(k)=\frac {I(k-1)}{z+2} +\frac {(1+z)^{1-k}-1}{(1-k)(z+2)}$$

If you now want then you can solve this recursion but I don't think that would be a great and easy job to do.

Hope it helps